Theorem im2anan9 587

Description: Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)

Hypotheses

Ref	Expression
im2an9.1	|- ( ph -> ( ps -> ch ) )
im2an9.2	|- ( th -> ( ta -> et ) )

Assertion

Ref	Expression
im2anan9	|- ( ( ph /\ th ) -> ( ( ps /\ ta ) -> ( ch /\ et ) ) )

Proof of Theorem im2anan9

Step	Hyp	Ref	Expression
1		im2an9.1	. . 3 |- ( ph -> ( ps -> ch ) )
2	1	adantr 274	. 2 |- ( ( ph /\ th ) -> ( ps -> ch ) )
3		im2an9.2	. . 3 |- ( th -> ( ta -> et ) )
4	3	adantl 275	. 2 |- ( ( ph /\ th ) -> ( ta -> et ) )
5	2, 4	anim12d 333	1 |- ( ( ph /\ th ) -> ( ( ps /\ ta ) -> ( ch /\ et ) ) )

Syntax hints:   -> wi 4   /\ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  im2anan9r  588  trin  4036  xpss12  4646  f1oun  5387  poxp  6129  brecop  6519  enq0sym  7240  genpdisj  7331  tgcl  12233  txlm  12448